What comes first here? pemdas doesnt really tell me what to do here

Question

So I have this equation: $2x(x+3)(x+3)$

Do I FOIL the $(x+3)$ first or multiply the $2x$ to the first $(x+3)$? Would there be a difference? Isn't multiplication commutative?

Answer 1

Multiplication is associative. That is, $a \cdot b \cdot c= (a \cdot b)\cdot c.$

So, for $2x(x+3)(x+3)$, we can find $2x(x+3)$ and multiply that by $(x+3).$

Elaborating further, we have $2x(x+3)(x+3)=(2x^2+6x)(x+3)=2x^3+12x^2+18x$

Furthermore, we could use PEMDAS here. There are no parentheses or exponents, so we simply multiply from left to right, the same way you would evaluate $2\cdot 3 \cdot 4.$

Answer 2

The expression $$2x(x+3)(x+3)$$ (its not an equation) can be expanded in a variety of ways. Perhaps the easiest is: $$2x(x+3)(x+3) = 2x(x+3)^2 = 2x(x^2+6x+9) = 2x^3+12x^2+18x$$

Whatever method you prefer is just fine, so long as it adheres to the laws of arithmetic.

Answer 3

Multiplication is associative. Hence, $(ab)c = a(bc)$. Therefore, the order in which the terms in the expression are multiplied does not matter.

To illustrate this observation with the product $2x(x + 3)(x + 3)$, we will show that first multiplying the first two terms of the expression, then multiplying the result by the third term yields the same result as first multiplying the last two terms of the expression, then multiplying the result by the first term. \begin{align*} 2x(x + 3)(x + 3) & = [2x(x + 3)](x + 3)\\ & = (2x^2 + 6x)(x + 3)\\ & = 2x^2(x + 3) + 6x(x + 3)\\ & = 2x^3 + 6x^2 + 6x^2 + 18x\\ & = 2x^3 + 12x^2 + 18x \end{align*} and \begin{align*} 2x(x + 3)(x + 3) & = 2x[(x + 3)(x + 3)]\\ & = 2x[x(x + 3) + 3(x + 3)]\\ & = 2x(x^2 + 3x + 3x + 9)\\ & = 2x(x^2 + 6x + 9)\\ & = 2x^3 + 12x^2 + 18x \end{align*}

Note that $2x(x + 3)(x + 3)$ is an algebraic expression, not an equation. The equation $2x(x + 3)(x + 3) = 2x^3 + 12x^2 + 18x$ asserts that the algebraic expressions $2x(x + 3)(x + 3)$ and $2x^3 + 12x^2 + 18x$ are equivalent.